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Learning about Distress
Christian C. G. Opp∗
April 8, 2015
Abstract
I develop an analytically tractable dynamic asset pricing model to study expected
returns of financially distressed firms in the presence of learning and investor activism.
Learning critically affects distressed stocks’ valuations and risk exposures as information about solvency is essential for firm survival in distress. Informational externalities
from active investors thus can also have first-order effects on distress risk premia. The
presented model can shed light on a variety of empirical regularities related to financial
distress, such as distressed firms’ apparent stock market underperformance, momentum return dynamics, and negative abnormal returns after private placements of public
equity involving active investors.
Keywords: Expected Returns, Financial Distress, Learning, Momentum, Active Investors, Risk Premia
∗
The Wharton School, University of Pennsylvania. Email: [email protected]. This paper was
previously circulated under the title “Learning, Active Investors, and the Returns of Financially Distressed
Firms.”I am grateful for comments and suggestions from Hui Chen, Alex Edmans, Dirk Hackbarth, Vincent
Glode, Alan Moreira, Paul Pfleiderer, Matt Spiegel, Ilya Strebulaev, Dimitri Vayanos, Jessica Wachter and
from seminar participants at Boston University, the Wharton School, Yale School of Management, UT
Austin, University of Notre Dame, Stanford GSB, London School of Economics, the WFAs, and the Kellogg
Junior Finance Conference. Research support from the Marshall Blume Prize in Financial Research, the
Rodney White Center for Financial Research, and the Wharton School Dean’s Research Fund is gratefully
acknowledged.
1.
Introduction
“After all, you only find out who is swimming naked when the tide goes out.”
Warren Buffett in the letter to shareholders (2001).
Financial distress is a central phenomenon in financial economics that has gained renewed
attention after the recent financial crisis and Great Recession. A key problem associated with
financial distress is investors’ challenging task to determine firm solvency: lacking precise
information, investors may be unable to distinguish solvent from insolvent firms and thus
be unwilling to provide finance when it is needed the most. The availability of information
and investors’ learning are thus essential for firm survival in distress, creating a feedback
effect on firm outcomes — investors’ learning not only reduces subjective uncertainty but
also affects a firm’s survival and thus, its actual riskiness.
In this paper I aim to shed light on the dynamic asset pricing implications of this learningbased feedback effect. I develop a continuous-time model of an indebted firm that, after
receiving a negative shock to earnings, attempts to raise external funds to avoid default.
Yet agents face uncertainty about the nature of the shock: earnings may be permanently
depressed, rendering the firm fundamentally insolvent. Investors make dynamic decisions
about capital injections and rationally update their beliefs about the firm’s future prospects
in the presence of business-cycle risks. Apart from small investors that learn passively from
public information I also consider large, active investors that can obtain superior information and assume a pivotal role in the firm’s access to finance. Despite the presence of both
aggregate risks and learning about firm-specific prospects the setting is tractable and allows
a parsimonious analysis of the optimal default boundaries. The model makes a large set of
novel empirical predictions that shed light on several asset pricing puzzles related to financial distress, such as distressed firms’ apparent stock market underperformance, momentum
return dynamics, and the connection between active investors and expected returns.1
In a setting that has similarities with standard EBIT-based capital structure models2
1
See, e.g., Dichev (1998), Campbell, Hilscher, and Szilagyi (2008), Avramov, Chordia, Jostova, and
Philipov (2007, 2013), and Park (2013). I discuss related empirical evidence in more detail later in the
introduction.
2
See, e.g., Goldstein, Ju, and Leland (2001), Hackbarth, Hennessy, and Leland (2007), and Strebulaev
(2007).
1
this paper considers a firm with a stochastic earnings process that may reach states where
the firm has to raise new equity to avoid defaulting on its existing perpetual debt. Investors
are, however, uncertain about the firm’s underlying quality, which is determined by a hidden
firm-specific Markov state that affects the firm’s future cash flow dynamics. Whereas good
firms have rising expected earnings and are fundamentally solvent, bad firms have declining
expected earnings and are truly insolvent. As investors cannot observe the hidden state
directly, capital provision depends critically on investors’ current beliefs and, in a dynamic
world, on expectations about the future speed of learning.
Even if investors currently have pessimistic beliefs about a firm’s prospects they may be
willing to provide additional funds if the speed of learning is high and precise information
about true firm quality is expected to arrive quickly. While waiting for more information
for a short period of time may require injecting additional funds to keep the firm afloat, it
gives equity investors an important benefit: default decisions can be delayed to a point in
time when more information is available, which reduces the propensity of triggering default
in error.
In contrast, supporting a distressed firm is less attractive when learning is slow. In
this case, investors need to inject capital for an extended period of time before they can
make a more informed default decision. However, the substantial capital they provide in the
meantime may be wasted on a truly insolvent firm and thus constitute an involuntary subsidy
to debt holders. Thus, holding current beliefs fixed, a slower speed of learning can render
equity investors unwilling to provide more funds in the first place and lead to immediate
default. As a result, equity holders’ default option value is highly sensitive to the speed of
learning, not just to current beliefs.3 As a firm’s default probability increases, learning speed
in turn becomes a more and more important determinant of the stock price.
This basic insight has important implications for risk premia of firms with high default
risk: risk exposures are affected not only by earnings risk and leverage but also by the
exposure of learning speed to aggregate fluctuations. Bayesian filtering based on observed
3
In this context it is important to distinguish investors’ conditional uncertainty about earnings prospects
from the unconditional physical uncertainty of earnings. Whereas increased physical uncertainty raises the
ex ante value of equity holders’ default option, a fast reduction in conditional uncertainty, that is, a better
ability to predict future earnings, also increases the ex ante option value, as it improves investors’ dynamic
financing and default decisions.
2
earnings performance in turn dictates that learning speed is higher when good and bad firms
have a higher propensity to exhibit differential earnings performance. In particular, if truly
insolvent firms exhibit large downside risks during the test of an aggregate downturn, learning
speed increases in these times and becomes counter-cyclical.4 Related to Schumpeter’s (1934)
hypothesis that recessions can have a “cleansing effect,” truly insolvent firms then default
more quickly in downturns, as their adverse performance separates them from good firms,
which show more resilience. Expectations about a faster separation support distressed firms’
stock prices in these times, as the highlighted learning-based feedback effect operates.
As a result, learning can have substantive implications for distressed firms’ asset prices:
exposures to business-cycle risk and risk premia can become negative, even when firms’ earnings processes are strongly pro-cyclical. If learning speed is not counter-cyclical but instead
constant or moderately pro-cyclical, equity risk exposures are still reduced in comparison to
the full information case. With increasing default risk, stock prices generically load more
and more heavily on the cyclical movements of learning speed rather than just earnings
risk. Learning may thus be a factor contributing toward the empirical finding that high
default risk is associated with low future stock returns (Dichev, 1998, Campbell, Hilscher,
and Szilagyi, 2008), a phenomenon referred to as the “distress anomaly.”5
Going beyond an explanation for low risk premia, the theory can further provide a coherent story for the observed negative unconditional alphas on portfolios consisting of the most
financially distressed stocks. When the quintile of stocks with the highest default risk has
higher average default risk in downturns, learning can reduce the quintile portfolio’s exposure to business-cycle risks more in these times (the learning-based feedback effect increases
with default risk). When market risk premia rise in downturns, these cyclical movements
4
See, e.g., Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (2014) for empirical evidence that
cross-sectional dispersion in productivity is strongly countercyclical.
5
Gao, Parsons, and Shen (2013) provide evidence on the robustness of the distress anomaly based on
a broad international data set of 39 countries. For US data Vassalou and Xing (2004) document that the
size effect in expected returns exists only in segments of the market with high default risk, and that this
is also largely the case for the book-to-market effect. Vassalou and Xing (2004) further find some evidence
that distressed stocks with a low distance to default have higher returns, but this evidence comes entirely
from small value stocks. Da and Gao (2010) further provide evidence that distressed firms’ stock returns in
Vassalou and Xing (2004) are biased upwards by 1-month reversal and bid-ask bounce. Griffin and Lemmon
(2002) document that among firms with the highest distress risk the difference in returns between high and
low book-to-market securities is more than twice as large as that in other firms. Further, the authors find
that firms with high distress risk exhibit the largest return reversals around earnings announcements.
3
imply a negative correlation between market risk premia and conditional betas of the quintile
portfolio. As a result, unconditional CAPM regressions yield upward biased beta estimates
for distressed firms, and negatively biased alpha estimates.6 The notion that conditional
betas can contribute toward resolving the distress anomaly is further supported by empirical
evidence in O’Doherty (2012) who finds that the correlation between market risk premia
and conditional betas of distressed stocks can account for about two thirds of the negative
unconditional alpha.
The same learning channel that dampens risk premia also implies that a momentum
strategy that goes long recent winners and shorts recent losers among distressed firms can
generate large expected returns. Positive correlations between past price changes and future
expected returns obtain because firms that experienced negative contemporaneous returns
have higher default risk, and thus, lower risk premia going forward, provided that learning is
sufficiently fast in downturns. Consistent with this mechanism, Avramov, Chordia, Jostova,
and Philipov (2007) find that both the extreme loser and winner portfolios consist of stocks
with the lowest and the next-lowest credit rating, respectively. The authors further find that
momentum profits are restricted to high credit risk firms and are nonexistent for firms of
high credit quality.7 Avramov, Chordia, Jostova, and Philipov (2013) further document the
importance of financial distress for the profitability of a variety of anomaly-based trading
strategies, such as, earnings momentum, dispersion, idiosyncratic volatility, and capital investments. In addition, consistent with the theory’s link between risk premia and exposures
to business cycle risks, Bansal, Dittmar, and Lundblad (2005) find that exposures to persistent changes in growth can explain a large fraction of the variation in expected returns
across momentum portfolios.
While in my baseline model small investors learn passively from realized earnings performance, I also analyze an extension of the model that features a large active investor.
6
See, e.g., Grant (1977) and Jagannathan and Wang (1996).
Bandarchuk and Hilscher (2013) find that existing strategies that yield elevated momentum profits
benefit from trading in stocks with more extreme past returns. High default risk and corresponding leverage
constitute a natural economic reason for extreme stock returns — in the cross-section, extreme past returns
are thus likely generated by firms with high default risk. Regressions reported in Bandarchuk and Hilscher
(2013) further indicate that credit ratings have a significant effect on momentum profits even after controlling
for idiosyncratic volatility and a measure of extreme past returns. Moreover, Avramov, Chordia, Jostova, and
Philipov (2013) find that the effect of idiosyncratic volatility on expected returns is explained by measures
of credit risk.
7
4
In practice, active investors, such as activist hedge funds or private equity funds, typically
acquire substantial stakes in distressed firms’ private placements of public equity.8 In the
model, these specialized investors can obtain additional information on a firm’s underlying
state. The firm issues equity to such an investor at a discounted price when it requires new
funds and small investors are uncertain about the firm’s solvency.9 Despite the discounted
offer, involving the active investor is in the interest of existing shareholders who can freeride by following the active investor’s lead going forward.10 As a result, the active investor
assumes a pivotal role in the firm’s access to finance, which is shown to create an externality affecting other investors’ risk premia. In particular, if learning based on a firm’s public
information is slow in aggregate downturns, the involvement of an active investor can help
off-load systematic risk from equity to debt holders.
This informational externality in turn can generate the empirical phenomena that (1)
financially distressed firms that issue discounted equity to new investors via private offerings
of public equity have particularly low expected returns, and (2) the buy-and-hold returns
after private offerings are negatively correlated with the discounts that were initially provided
to the large investors that participated in these offerings.11 In the model, the firm is willing
to offer larger discounts to active investors with greater skill. A larger discount thus forecasts
more effective involvement of the active investor going forward, and thus a stronger effect on
equity risk exposures. In other words, the involvement of an active investor is a state variable
that affects distressed firms’ conditional expected returns. The highlighted informational
8
See, e.g., Park (2013) for related empirical evidence.
Providing one investor with a large equity stake can help resolve free-rider problems present among small
investors (see, e.g., Grossman and Hart (1980) and Shleifer and Vishny (1986)).
10
Consistent with the model, Brophy, Ouimet, and Sialm (2009) document that hedge funds tend to finance
companies that have poor fundamentals and pronounced informational frictions, and require substantial
discounts.
11
Park (2013) documents these empirical facts. Several other empirical papers analyze the relation between
private placements and equity returns, and find related results. Krishnamurthy, Spindt, Subramaniam, and
Woidtke (2005) find that shareholders not participating in private placements experience post-issue negative
long-term abnormal returns. Hertzel, Lemmon, Linck, and Rees (2002) also document that public firms
that place equity privately experience negative post-announcement stock-price performance. Hertzel and
Smith (1993) provide empirical evidence that discounts provided in private placements reflect information
costs borne by private investors, which is consistent with the mechanism in my model. Brophy, Ouimet,
and Sialm (2009) also find that public companies that raise equity privately from large specialized investors
such as hedge funds significantly underperform companies that obtain financing from other investors in the
future.
9
5
externality can also shed some light on the intriguing empirical finding by Gao, Parsons,
and Shen (2013) that the distress anomaly is especially strong among small stocks in North
America and Europe, markets where distressed equity hedge funds are particularly active.12
Related literature. This paper is generally related to a growing literature on learning
in financial markets13 and the relationship between financing decisions and asset pricing.14
Business-cycle risks of the type considered in this paper play a prominent role in dynamic
capital structure models, such as Chen (2010) and Bhamra, Kuehn, and Strebulaev (2010).
A paper related to mine that integrates both credit risk and learning in a dynamic economy
is David (2008). Whereas in David (2008) agents learn about the hidden drifts of aggregate
processes (such as real earnings and inflation), agents in my model learn about firm-specific
states that determine future earnings dynamics. Critical for the results I obtain is the fact
that I can solve for equity holders’ true endogenous default boundaries. In contrast, David
(2008) makes exogenous assumptions on default decisions in order to solve for prices. In
particular, by assuming the same default rules as in Merton (1974), David’s (2008) approach
does not capture equity holders’ true option value.
My paper’s main objective is to provide a tractable model that illustrates the effects of
learning and investor activism on financially distressed firms’ risk premia dynamics. I do
not argue that the highlighted learning channel is the only plausible mechanism that can
help explain the discussed empirical regularities, such as the distress anomaly. For example,
investor irrationality or additional institutional frictions can certainly be contributing factors
for these phenomena. Yet agents’ learning about firm solvency is an economic force that is
generically relevant in financial distress and appears to warrant a rigorous formal analysis.
Regarding low returns for distressed firms, complementary theoretical explanations have
been put forward that have other virtues and limitations. In contrast to my paper, these
other theories do not analyze the effects of learning and investor activism on distress risk
12
Since these are cross-country correlations several alternative explanations exist. For example, Gao,
Parsons, and Shen (2013) interpret this correlation as consistent with investor overconfidence, a behavioral
bias that could be more prevalent in the cultures of North America and Europe, which stress individualism.
13
See Pastor and Veronesi (2009) for a survey of this literature.
14
See, e.g., Hackbarth, Miao, and Morellec (2006), Livdan, Sapriza, and Zhang (2009), and Gomes and
Schmid (2010).
6
premia and generally cannot produce negative expected returns. My paper further also
provides theoretical reasons why distress might not only be associated with low returns
but also negative alphas. George and Hwang (2010) argue that firms with high financial
distress costs choose low leverage to avoid distress but retain exposure to the systematic risk
of bearing such costs in low states, implying that they have higher expected returns than
highly levered firms. Garlappi and Yan (2011) provide a model that shows how potential
shareholder recovery upon resolution of financial distress (violation of the absolute priority
rule) may effectively imply de-levering upon default, which can lead to lower expected returns
for firms with high default probabilities. Similarly, Garlappi, Shu, and Yan (2008) argue that
bargaining between equity holders and debt holders in default may account for low expected
equity returns for firms with high default risk, given that shareholders can extract high
benefits from renegotiation.15 McQuade (2013) provides a model where aggregate volatility
risk commands a separate, negative market price and financially distressed stocks, due to
their optionality, have lower volatility risk premia than financially healthy firms. Ozdagli
(2013) emphasizes that risks under the physical and risk-neutral measure do not need to
coincide, implying that the equity of firms with high physical default risk does not need to
command high risk premia. Consistent with this idea, I provide a learning-based economic
rationale for why financially distressed firms should indeed have low exposures to priced
risk despite their high physical default risk. Finally, unrelated to financial distress, Johnson
(2002) provides an alternative rational explanation of momentum effects based on stochastic
expected growth rates.
In the following, I first present the baseline model that features passive learning from
firms’ earnings performance. In section 3, I extend this setup to incorporate an active
investor. Section 4 concludes.
15
Gao, Parsons, and Shen (2013) shed some light on the role of this channel by analyzing international
data. Based on a broad data set of 39 countries the authors find that the distress anomaly is not related to a
country’s creditor protection environment. Hackbarth, Haselmann, and Schoenherr (2013) provide evidence
that weaker creditor rights can be associated with lower equity risk premia for distressed firms. However,
the authors acknowledge that this finding does not resolve the distress anomaly, since it cannot explain why
the CAPM and the three factor model yield negative alphas for financially distressed firms.
7
2.
The Baseline Model
I consider a continuous time economy with exogenously specified aggregate dynamics and
analyze asset pricing implications for marginal firms in the presence of learning about firm
solvency.
2.1.
Aggregate States and Stochastic Discount Factor
The aggregate state of the economy, denoted by Z, follows a two-state continuous time
Markov chain with Z ∈ ΩZ = {G, B}, where G refers to a high-growth state (boom) and
B refers to a low-growth state (recession). All agents can observe this aggregate state. I
denote the transition rate between state Z and Z 0 by λ (Z, Z 0 ), where Z 0 6= Z. The stochastic
discount factor (SDF) follows a Markov-modulated jump process,16
dξt
0
= −rf (Zt− ) dt + (eφ(Zt− ,Z ) − 1) (dNt (Zt− , Z 0 ) − λ(Zt− , Z 0 )dt) ,
ξt−
(1)
where rf (Z) is the risk-free rate, Nt (Z, Z 0 ) is a counting process that keeps track of the
number of Markov chain jumps from state Z to state Z 0 , and φ (Z, Z 0 ) represents the jump
risk premia.17
The postulated SDF dynamics obtain for example in a Lucas-Breeden economy where
the drift of aggregate consumption is governed by a continuous time Markov chain and
agents have Duffie and Epstein (1992a,b) preferences. While sharing Markov chain dynamics
with the SDFs in several recent credit risk models18 this SDF deliberately abstracts from
Brownian shocks, a key technical choice that ensures analytical tractability in my setting,
which features learning. By considering a larger number of Markov states the model can
in principle still capture rich dynamics and provide good approximations to more complex
processes. For the purposes of this paper, however, I will focus on the case of two aggregate
16
In the following, all processes will be right continuous with left limits. Given a process yt , the notation
yt− will denote lims↑t ys , whereas yt denotes lims↓t ys .
0
17
Thus, the jump intensity between states Z and Z 0 under the risk neutral measure is eφ(Z,Z ) λ(Z, Z 0 ).
18
See, e.g., Chen (2010), Bhamra, Kuehn, and Strebulaev (2010), Chen, Xu, and Yang (2012), Chen, Cui,
He, and Milbradt (2014).
8
states, which suffices to highlight the central insights and increases the transparency of the
results.
2.2.
The Firm
I follow standard EBIT-based capital structure models19 in that the earnings of a firm are
split between a coupon promised to debt holders and a dividend paid to equity holders. Yet,
since my paper’s focus is not on capital structure but the asset pricing implications of learning
conditional on reaching distress, I take it as given that the firm issued debt in the past. For
this reason, I also abstract from taxes and other frictions that might generate incentives to
issue additional debt. Further, to set the channel apart from existing papers on the effects
of violations of absolute priority, I assume that equity holders cannot renegotiate contracts
with existing debt holders.20 An alternative interpretation of this setup is that required
coupon payments represent operational cost that the firm has to pay to avoid defaulting on
suppliers, customers, or employees. As the leading narrative, I will, however, refer to the
receivers of coupon payments as debt holders. Throughout, management acts in the interest
of shareholders. Further, in the baseline model, all agents have symmetric but potentially
incomplete information.
At time t = 0, the firm has obligations that promise debt holders a constant perpetual
coupon rate c. The firm’s before-interest earnings rates X (z) ≥ 0 are governed by firmspecific Markov states z ∈ Ωz the dynamics of which depend on the aggregate state Z ∈
ΩZ . Specifically, the transition rate from earnings state z to state z 0 is given by λz,z0 (Z).
By conditioning the transition rates on the aggregate state Z the setup can capture rich
dependencies between the firm’s earnings dynamics and aggregate conditions. Let Ω =
ΩZ × Ωz denote the complete set of states for the vector (z, Z).
To simplify the exposition, I will consider a particular stylized Markov state setup where
I divide the set of all states Ω into four disjoint subsets, to which I refer as initial solvent
states Ωs , opaque distressed states Ωd , and good and bad revealing sets of states Ωg and
Ωb , respectively. I discuss these subsets of states in more detail below. Figure I further
19
20
See, e.g., Goldstein, Ju, and Leland (2001), Hackbarth, Hennessy, and Leland (2007), Strebulaev (2007).
See, e.g., Garlappi, Shu, and Yan (2008), Garlappi and Yan (2011).
9
illustrates the setup.
Initial solvent states. At date t = 0, the firm is assumed to be in solvent states Ωs in
which it generates sufficiently high earnings to cover its interest expenses, that is, X (z) > c.
In these states the firm pays positive dividends to shareholders and is thus unambiguously
solvent.
Opaque distressed states. The firm can, however, receive negative shocks that push its
earnings below the coupon rate c. In particular, the firm can transition into opaque distressed
states Ωd = {dg, db} × {G, B}, where earnings are given by Xd < c. In these states, the firm
has to raise external funds in order to avoid default, but agents have incomplete information
about the firm’s prospects, as they cannot directly observe whether a benign (transitory) or
disastrous (permanent) shock hit.21
Earnings do not directly reveal the underlying firm-state z ∈ {dg, db} since X(z) = Xd
∀(z, Z) ∈ Ωd . After a benign shock, the firm is in a good firm-state, z = dg, where earnings
are likely to recover quickly. After a disastrous shock, on the other hand, the firm is in a bad
firm-state, z = db, where earnings are permanently depressed and recovery is impossible. To
simplify the exposition I will refer to firms in states dg and db as “dg-firms” and “db-firms”
respectively. These distressed states are defined such that, the firm is solvent in state dg
but insolvent instate db, that is, if equity investors could observe the true firm-state z, they
would optimally choose to default immediately in firm-state db and provide new funds in
firm-state dg.22 Depending on investors’ beliefs about the underlying state, default thus can
occur in the set of states Ωd . All investors are Bayesian learners and form rational beliefs
about the underlying state based on available information, such as the passage time in the
distressed state and other potential signals.
21
Here, a disastrous shock affects a single firm, although the arrival intensities of these shocks are allowed
to depend on the aggregate state Z. See, e.g., Rietz (1988), Barro (2009), Wachter (2013) for papers on the
asset pricing implications of disasters affecting aggregate consumption dynamics.
22
Investors’ decision problem would be degenerate if the equity value in firm-state dg was also non-positive,
since in that case the distressed firm would be always insolvent.
10
Initial solvent states
ΩΩs
Aggregate
state G (z,G)
Aggregate (z,B)
state B
Opaque distressed states
ΩΩd
Hidden firm-state z ∈ {dg,db}
(dg,G)
(db,G)
Revealing states
ΩΩg & ΩΩb (z,G) ∈ ΩΩg (z,G) ∈ ΩΩb (dg,B)
(z,B) ∈ ΩΩg (db,B)
(z,B) ∈ ΩΩb FIGURE I
The figure illustrates the Markov states that govern the firm’s earnings dynamics and aggregate conditions
(Ωs , Ωd , Ωg , and Ωb ). Depending on agents beliefs, the firm may default in opaque distressed states Ωd .
Revealing states. Depending on whether the true distressed firm-state is good or bad (dg
or db), the firm can transition into a good or a bad set of revealing states, denoted by Ωg and
Ωb respectively. The sets Ωg and Ωb are absorbing in the sense that once the firm reaches one
of these sets, it will stay in this set with probability one. This assumption by itself does not
put any restrictions on earnings dynamics as it does not restrict how the sets Ωg and Ωb are
constructed. Yet, to ensure consistency with the above-mentioned defining properties of the
distressed states dg and db, I restrict earnings in the sets Ωg and Ωb as follows: while earnings
in the good set, Ωg , are above the required coupon payments, implying solvency, earnings in
the bad set, Ωb , are below Xd , implying insolvency. Thus, when the firm’s earnings recover
or deteriorate further relative to Xd , agents can infer whether the firm was previously hit by
a benign or a disastrous shock.23
23
These clear-cut differences in earnings levels across the sets Ωg and Ωb are not essential for the results
but yield tractability. For example, one could add noise by assuming that the set Ωb also contains states
where earnings are above Xd . Yet, to be consistent with the definition of db as a firm-state where the firm is
11
Discussion: Markov state dynamics. To illustrate results in the most parsimonious
way I consider a Markov state setup where current states are observable until the firm is hit
by a negative shock that causes earnings to drop below the required coupon payments. While,
more generally, the underlying firm-state could already be unobservable before earnings drop
below the coupon, the firm would still be unambiguously solvent at that time, as dividends
are locally strictly positive. In other words, uncertainty about the hidden firm-state could
not raise doubts about solvency unless earnings also fall below the required coupon c. Once
this happens, equity holders would again form beliefs about the hidden state and evaluate
solvency, just as captured in the considered Markov state setup.
2.3.
Analysis
Solving for equity values backwards, I first analyze the revealing sets of states Ωb and
Ωg , then opaque distressed states Ωd , and finally the initial solvent states Ωs . The decision
problem at the heart of the analysis arises in opaque distressed states, where agents are
generally uncertain about the firm’s fundamental solvency.
2.3.1.
Revealing States
In the long-run, the firm either has defaulted while being in a opaque distressed state Ωd ,
or has reached a state in the revealing sets Ωb or Ωg . When the firm reaches a state in the
bad revealing set Ωb , equity holders optimally choose to default immediately, since the firm
is known to be insolvent. Equity values in the good revealing states Ωg are characterized in
the following proposition.
PROPOSITION 1 (Equity value in good revealing states). In good revealing states (z, Z) ∈
Ωg the firm’s equity value is given by
Z
V (zt , Zt ) = Et
t
∞
ξ (Zτ )
(X (zτ ) − c) dτ .
ξ (Zt )
(2)
insolvent, the probability that earnings recover and rise above the coupon rate c would have to be sufficiently
small.
12
The corresponding Hamilton-Jacobi-Bellman equation implies that the function V (z, Z) solves
the following system of linear equations for all (z, Z) ∈ Ωg :
X
0 = X (z) − c − rf (Z) V (z, Z) +
λzz0 (Z) (V (z 0 , Z) − V (z, Z))
z 0 ∈Ωz
φ(Z,Z 0 )
+λ (Z, Z 0 ) e
(V (z, Z 0 ) − V (z, Z)) .
(3)
The firm’s state-contingent equity risk premium rp (z, Z) is given by:
0
φ(Z,Z 0 )
rp (z, Z) = −λ (Z, Z ) e
V (z, Z 0 )
−1 .
−1
V (z, Z)
(4)
Proof. The result follows immediately from the Hamilton-Jacobi-Bellman equation. Since,
by assumption, X(z) > c for all (z, Z) ∈ Ωg , the firm does not default in the set Ωg .
The formula for a firm’s equity risk premium provided in equation (4) reflects the intuitive
notion that if stock prices V are positively exposed to business cycle risks, that is, changes
in the macro state Z, stocks command a positive risk premium.
2.3.2.
Opaque Distressed States
In opaque distressed states Ωd investors are generally uncertain about the underlying firmstate z ∈ {db, dg}. Let πt denote the probability under agents’ filtration Ft that the firm is
fundamentally solvent, that is, πt ≡ Pr [z = dg|Ft ]. The following Lemma characterizes the
evolution of posterior beliefs.
LEMMA 1 (Posterior beliefs). If at time t˜ the firm jumps from a solvent firm-state z = s
to a state with distressed earnings Xd , investors assign the following probability to the event
that the firm is fundamentally solvent (z = dg):
πt˜ =
λs,dg (Zt˜− )
.
λs,dg (Zt˜− ) + λs,db (Zt˜− )
13
(5)
For t > t˜, posterior beliefs πt evolve as follows:
dπt = θ (Z) (1 − πt ) πt dt − πt dNtdb + (1 − πt ) dNtdg ,
(6)
where θ (Z) is defined as:
θ (Z) =
X
λdb,z0 (Z) −
(z 0 ,Z)∈Ωb
X
λdg,z0 (Z) ,
(7)
(z 0 ,Z)∈Ωg
and where Ntz for z ∈ {dg, db} is a counting process that switches from 0 to 1 when the
firm jumps from an opaque distressed state z ∈ {dg, db} to the corresponding revealing set of
states (Ωg or Ωb ).24
Proof. The results follow from Bayes’ law and the martingale property of π.
θ (Z) in equation (7) captures the relative propensity with which good and bad firms
show improving and deteriorating earnings performance, respectively. Correspondingly, θ(Z)
determines whether beliefs are revised upwards or downwards as a function of passage time
without changes in earnings. θ(Z) generally depends on the current state of the aggregate
economy. For example, θ(G) takes a negative value if during economic booms (Z = G) good
firms’ earnings are more likely to recover than bad firms’ earnings are to deteriorate further.
Passage time without earnings changes is then regarded as negative news during booms and
thus leads to downward revisions in posterior beliefs πt . On the other hand, θ(B) takes a
positive value if, during economic downturns (Z = B), bad firms’ earnings are more likely
to fall further than good firms’ earnings are to recover. Correspondingly, constant earnings
are viewed as a sign of resilience in recessions, leading to improvements in posterior beliefs
πt .25
Based on the belief dynamics described in Lemma 1 I can now characterize the firm’s
equity value in opaque distressed states. I introduce the superscript R to indicate that this
equity value will also constitute a reservation value in the extended model that features an
24
After a jump to the good revealing set Ωg the probability that the firm was previously hit by a benign
shock jumps to one (that is, π = 1). In contrast, after a jump to the bad revealing set Ωb this probability
jumps to zero (that is, π = 0).
25
In the special case where θ(Z) = 0 passage time alone does not alter beliefs πt .
14
active investor (introduced in Section 3).
PROPOSITION 2 (Equity values in opaque distressed states). The firm’s equity value in
opaque distressed states is given by
Z
R
V (πt , Zt ) = max
Et
∗
{τ }
t
τ∗
ξ (Zτ )
(X (zτ ) − c) dτ = v R (πt , Zt )+ .
ξ (Zt )
(8)
The corresponding Hamilton-Jacobi-Bellman equation yields the following set of ODEs that
the function v R (π, Z) solves:
0 = Xd − c − rf (Z) v R (π, Z) + vπR (π, Z)θ (Z) (1 − π) π
X
πλdg,z0 (Z) (V (z 0 , Z) − v R (π, Z))
+
z 0 6=dg
−
X
(1 − π) λdb,z0 (Z) v R (π, Z)
z 0 6=db
0
+λ (Z, Z 0 ) eφ(Z,Z ) v R (π, Z 0 ) 1{π≥πR (Z 0 )} − v R (π, Z) .
(9)
Equity holders optimally default in aggregate state Z when posterior beliefs π fall below an
endogenous cutoff value π R (Z). In states Z where θ (Z) < 0 the following conditions apply:
0 = vπR (π R (Z) , Z) and 0 = v R (π R (Z) , Z). In states Z where θ (Z) > 0, the following
conditions apply: v R (π R (Z) , Z) = 0 and limπ↑1 v R (π, Z) = V R (dg, Z), where V R (dg, Z)
is characterized in Appendix A.2. If θ (Z) = 0 then the ODE for state Z simplifies to a
nonlinear equation.
Proof. See Appendix A.2.
Equity holders’ optimal default strategy can be summarized by the state-contingent belief
thresholds π R (Z). Investors provide additional equity as long as beliefs are above these
thresholds and let the firm default otherwise.
2.3.3.
Initial Solvent States
Finally, equity values in initial solvent states Ωs can be characterized analogously to those
in states Ωg and Ωd (see Propositions 1 and 2). To economize on space I relegate this part
15
of the analysis to Appendix A.1.
2.4.
A Calibration of the Baseline Model
In this section I parameterize the model to illustrate the asset pricing implications of
learning when firms are in distressed states. I discuss a baseline calibration as well as
comparative statics that illustrate the impact of alternative parameter choices. Parameter
values of the baseline calibration are listed in Table 1.
2.4.1.
Choosing Parameters
Stochastic discount factor. I use the values from Chen, Xu, and Yang (2012) and Chen,
Cui, He, and Milbradt (2014) to calibrate the parameters of the aggregate economy.26 According to this calibration, the risk-free rate is set to be constant at 5%. Transition rates
between aggregate states G and B are set such that expansions and recessions last, on average, 10 years and 2 years, respectively. Finally, the jump risk premium is chosen so that
risk premia are consistent with a Markov-modulated long-run risk model that is calibrated
to match the equity premium.27
Opaque distressed states Ωd . I normalize coupon payments c to one and calibrate the
shortfall in earnings in opaque distressed states Ωd using statistics provided in Campbell,
Hilscher, and Szilagyi (2008). To capture the notion that firms are distressed in states Ωd ,
I calibrate the parameter Xd to match a ratio of net income to market value of assets of
approximately −2.5%, which is representative for firms that have high default risk but do
not file for bankruptcy with certainty. This value is half a standard deviation above the
mean value for firms in Campbell, Hilscher, and Szilagyi’s (2008) “Bankruptcy Group” and
26
As discussed in Section 2.1, the SDF in my model deviates from the one in Chen, Xu, and Yang (2012)
and Chen, Cui, He, and Milbradt (2014) in that it does not feature Brownian shocks. The parameters
corresponding to Brownian shocks are thus also not part of my calibration.
27
For a consumption process with a growth drift of +0.05 in booms and −0.01 in recessions and a leverage
parameter for dividends of 3.5 (as in Bansal and Yaron, 2004) this SDF generates an average equity risk
premium of 6%.
16
about one standard deviation below the average value across all firms.28 Using this ratio,
I back out an earnings shortfall of 21% in opaque distressed states, which corresponds to
choosing Xd = 0.79.29
Further, consistent with the description of opaque distressed states in Section 2.2, earnings evolve according to a Markov-modulated jump process that depends on the underlying
hidden firm state:
dXt
= (eδz − 1)dNtz , for z ∈ {db, dg}.
Xt−
(10)
A log-change in earnings of size δz occurs when the counting process N z jumps and the firm
moves from the hidden state z ∈ {db, dg} to a revealing set of states, that is, Ωb or Ωg .
These earnings changes occur with Poisson intensities λdb,b (Z) and λdg,g (Z), respectively. In
the baseline calibration I choose symmetric values for δz , that is, δdg = −δdb = δ. Below, I
further discuss the extent to which asymmetric values for δz would affect the results.
Revealing sets of states Ωg and Ωb . I calibrate δ and the price-dividend ratios after recovery from distress (after a jump to Ωg ) so that the average price-dividend ratio and equity
risk premium match those of the market portfolio, as estimated by Bansal and Yaron (2004,
Table IV). The mean price-dividend ratio is 26.56 and the average equity risk premium is
6.33%. After recovery from distress, a firm’s key asset pricing moments are thus representative of the average firm in the market. Given the paper’s focus on distress it is convenient
to calibrate price-dividend ratios in post-recovery states Ωg directly, as they encode all information about dynamics in states Ωg relevant for equity valuations in opaque distressed
28
Campbell, Hilscher, and Szilagyi (2008) find that the average ratio of net income over market value of
total assets is −4% for firms in the “Bankruptcy Group,” which is defined as the sample of firms in months
immediately preceding a bankruptcy filing. The standard deviation of this ratio in the Bankruptcy Group is
3%, implying that I calibrate to a ratio of −2.5%. This number (−2.5%) is also approximately one standard
deviation below the average ratio of net income over market value of assets in the total sample.
29
Using a recovery rate of 41.3% and approximating the market value of assets as the recovery value of debt
Xd −c
1
= 20, one obtains the relation 0.413·20
= −0.025,
based on a promised risk-free perpetuity value of rcf = 0.05
or equivalently, Xd = 1−0.025·20·0.413 = 0.7935. The recovery rate of 41.3% is the long-term mean recovery
rate based on data provided in Moody’s “Corporate Default and Recovery Rates, 1920-2008,” which is also
documented in Chen (2010).
17
Table 1
Baseline Parameters. The table lists parameter values of the baseline calibration. The transition density between aggregate states λ(Z, Z 0 ), the jump risk premium exp(φ), and the risk-free rate rf , are taken
from Chen, Xu, and Yang (2012) and Chen, Cui, He, and Milbradt (2014). Coupon payments c are normalized to one. Earnings in opaque distressed states, Xd , are chosen to match a ratio of net income to
market value of assets of approximately −2.5%, which is representative of firms with high default risk
(see summary statistics in Campbell, Hilscher, and Szilagyi, 2008). The parameter δ and price-dividend
ratios after a jump into the recovery set Ωg are calibrated to match the equity risk premium and the
average price-dividend ratio of the market portfolio, as estimated by Bansal and Yaron (2004). Poisson
intensities for dg-firms are chosen to match average earnings volatility of 30% and an average equity risk
premium of 6.33%, consistent with the one after a jump to the recovery set Ωg . Poisson intensities for
db-firms are chosen to match an average earnings volatility of 30% and average systematic volatility of
10% at a one-year horizon. These earnings volatilities are chosen to be consistent with calibrations in
the existing literature (e.g., Bhamra, Kuehn, and Strebulaev, 2010, Chen, Cui, He, and Milbradt, 2014).
Parameter
Aggregate Economy
Transition rates for aggregate states
Jump-risk premium
Risk-free rate
Firm
Earnings in opaque distressed states
Size of log-earnings changes
P-D ratio after recovery (jump to Ωg )
Poisson intensity of recovery by dg-firm
Poisson intensity of decline by db-firm
Variable
λ
exp(φ)
rf
State G State B
0.1
2.0
0.5
0.5
0.05
Xd
δ
0.79
0.30
V
X−c
28.51
1.10
0.70
λdg,g
λdb,b
16.80
0.50
2.50
Source
Chen, Xu, and Yang (2012)
Chen, Xu, and Yang (2012)
Chen, Xu, and Yang (2012)
Literature;
Literature;
Literature;
Literature;
Literature;
see
see
see
see
see
section
section
section
section
section
2.4.1
2.4.1
2.4.1
2.4.1
2.4.1
states Ωd .30 Moreover, since, by definition, equity prices are zero after a jump into the bad
revealing set Ωb , no additional structure needs to be imposed on earnings in this set.31
Poisson intensities for dg-firms. The level and variation of Poisson intensities λdg,g
across the two aggregate states G and B are chosen to match an unconditional earnings
30
The parameter choices Xd = 0.79, δ = 0.3, and c = 1, determine the level of dividends immediately after
recovery (after a jump to Ωg ). Dividends and P-D ratios pin down equity values. As shown in Proposition
2, equity values in distressed states Ωd can be characterized as a function of the average equity value after
recovery — all relevant properties of earnings dynamics in the set of states Ωg are encoded in these prices.
31
Details about earnings dynamics in the set Ωb do not affect equity valuations in distressed states Ωd ,
since, by definition, earnings prospects in the set Ωb are so poor that immediate default is optimal.
18
volatility of 30%32 and an average equity risk premium of 6.33% for firms that are believed
to be a dg-firm with probability one.33 Recent papers that calibrate structural credit risk
models feature similar earnings volatility.34 Note that to match a risk premium of 6.33%
earnings have to be calibrated to be procyclical — as a result, dg-firms are more likely to
recover during aggregate booms than during recessions.
Poisson intensities for db-firms. The Poisson intensities λdb,b are chosen to match an
average earnings volatility of 30%35 and an average systematic volatility of 10% at a oneyear horizon.36 As in the case of dg-firms, these numbers are broadly consistent with choices
made in other papers in the literature.37 Since the firm is insolvent for π → 0 and thus does
not have a well-defined equity risk premium, I target systematic earnings volatility in the
case of db-firms. The corresponding parameter choices imply strongly pro-cyclical earnings
dynamics for db-firms: on average, a db-firm exhibits earnings declines after 7 months if the
economy is currently in a recession; this number is 16 months for a boom.38
Asymmetry of business cycles. Since recessions are short-lived relative to aggregate
booms (unconditional probability of 1/6 vs. 5/6, respectively) the calibration is inherently
asymmetric. Thus, when db-firms and dg-firms have the same average earnings volatility
and pro-cyclical earnings, their Poisson intensities are naturally not completely symmetric
32
That is, average local earnings volatility of a dg-firm is given by E[Std[d log X|Z]|dg] = 0.30dt.
That is, limπ↑1 E[rp(π, Z)] = 6.33%.
34
For example, in Chen, Cui, He, and Milbradt (2014) local idiosyncratic volatility is 22.5% and local
systematic volatility is on average 10.8%. In Bhamra, Kuehn, and Strebulaev (2010) these numbers are
22.6% and 10.1%, respectively. Both models additionally feature pro-cyclical variation in earnings growth
drifts, which yields additional variation at lower frequencies.
35
That is, average local earnings volatility of a db-firm is given by E[Std[d log X|Z]|db] = 0.30dt.
36
Whereas instantaneous earnings risk is purely idiosyncratic, cumulative earnings changes are correlated
with the aggregate conditions since the Poisson intensities λdb,b vary with the aggregate state Z. Let τG
R t+1
denote the cumulative time spent in aggregate state Z = G over the next year, that is, τG = t 1{Zs =G} ds
and let ∆x = log (Xt+1 /Xt ). I compute the average of systematic volatility Stdt [Et [∆x|τG ]] across aggregate
states Zt ∈ {G, B}.
37
As noted in Section 2, the specifications for earnings dynamics in Bhamra, Kuehn, and Strebulaev (2010)
and Chen, Cui, He, and Milbradt (2014) are not identical to the ones postulated in this paper, as they feature
diffusion risk and abstract from learning about hidden states. See also footnote 34.
38
These expected arrival times account for mean-reversion in the aggregate state Z. Formally, I compute
R τ?
the conditional expectation E[ t τ dNτdb dτ |Zt ] for Zt ∈ {G, B}, where τ ? denotes the first time after date t
that the counting process Ntdb jumps.
33
19
either. While the intensity of an earnings change is highest for db-firms in recessions, the
average arrival time of earnings news (across all Z) is actually longer for db-firms (14 months)
than for dg-firms (13 months). This difference arises because booms tend to last longer and
dg-firms have a high intensity of recovery during these times.
In unreported calibrations I confirm that in an economy with symmetric business cycles
qualitatively similar results can obtain when dg-firms’ and db-firms’ Poisson intensities are
symmetric mirror images of each other.39 However, since the asymmetry of business cycles
is an empirically robust phenomenon, it is also a feature of the baseline calibration.40
2.4.2.
Results of the Calibration
The main results of the calibration can be best understood by studying the dependence
of equilibrium equity prices on the two state variables, aggregate conditions Z and agents’
beliefs about the firm’s fundamental solvency π. The left-hand side panel of Figure II plots
stock valuations as a function of these two states in a region close to the default boundaries.
The right-hand side panel plots equity risk premia. The two lines in each panel refer to the
two aggregate states, G and B.
Belief dynamics. The calibration implies the intuitive notion that distressed firms that
show resilience in downturns by maintaining stable earnings exhibit upward revisions in
posterior beliefs over time. In other words, in a downturn, passage time without earnings
changes improves agents’ posterior beliefs and correspondingly leads to an increase in equity
values.41 In contrast, in booms, stable earnings are considered a bad signal, since truly
solvent firms are likely to recover quickly when general economic conditions are good. In
booms, passage time without earnings changes is thus a sign of weakness and associated
with deteriorating beliefs and declining equity values. Referring to the left-hand side panel
of Figure II, in recessions (B), valuations thus drift upwards on the equity value function
39
That is, a parameterization where booms and recessions have equal probability (λ(G, B) = λ(B, G)),
and where dg-firms and db-firms have pro-cyclical intensities that are mirror images of each other, i.e.,
λdg,g (B) = λdb,b (G) and λdg,g (G) = λdb,b (B).
40
See, e.g., Bloom (2009), Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (2014), Kehrig (2015).
41
Statements about the evolution of beliefs as a function of passage time refer to the evolution of beliefs
absent earnings changes. Unconditionally, beliefs are a martingale.
20
absent earnings changes. In contrast, in booms (G), valuations drift downwards absent
earnings changes, and smoothly approach zero, where equity holders trigger default.42
Equity valuations. Equity valuations in both aggregate states are naturally increasing in
beliefs π. For optimistic beliefs (high π) stock valuations are pro-cylical, that is, for given
beliefs π, they are higher in booms (G) than in recessions (B). Yet the opposite is true in a
region close to the default boundaries. Here, the equity value is lower in booms. This is true
despite the fact that earnings are highly pro-cyclical in the calibration. As I will explain
below, endogenous default decisions informed by learning generate this surprising result. As
a benchmark it is useful to keep in mind that pro-cyclicality of earnings ensures that the
unlevered firm would always have higher valuations in booms than in recessions.43
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FIGURE II
The graphs plot the firm’s equity values (left-hand side panel) and equity risk premia (right-hand side panel)
as a function of the posterior probability that the firm is fundamentally solvent, πt = Pr [z = dg|Ft ], and the
aggregate state Zt ∈ {G, B}. The plot of equity values focuses on a region close to the default boundaries to
highlight the behavior of stock prices when default risk is high. All parameter values are provided in Table 1.
42
Given the considered parameter choices, smooth pasting only applies in state Z = G, not in state Z = B.
In state Z = B, passage time causes beliefs to drift upwards, since db-firms are more likely to exhibit a decline
in recessions than dg-firms are to exhibit a recovery.
43
The unlevered firm’s equity value is given by the probability weighted average of the equity values in state
dg and db, that is, VU (π, Z) = (1 − π)VU (db, Z) + πVU (dg, Z), where the subscript U indicates the unlevered
equity value. Pro-cyclicality of earnings implies that VU (db, G) > VU (db, B) and VU (dg, G) > VU (dg, B),
such that VU (π, G) > VU (π, B) for all π.
21
Risk premia and momentum. The described inversion of valuations close to the default
boundaries has striking implications for firms with high default risk: equity risk premia turn
negative, as shown in the right-hand side panel of Figure II. Whereas risk premia are high
for optimistic beliefs — for π = 1 the average risk premium is 6.33% — they turn negative
when beliefs become sufficiently pessimistic.
As a result, the model can generate momentum dynamics for distressed firms. Negative
updating about firm quality leads not only to negative contemporaneous returns, but also to
lower expected equity returns going forward, as the firm gets closer to the default boundary.
“Recent losers” thus have lower expected returns going forward. Conversely, good news that
is associated with positive price changes pushes the firm away from the default boundary
and leads to higher future expected returns for “recent winners.” A momentum strategy
that goes long recent winners and shorts recent losers can therefore generate a large positive
spread in expected returns. While the results reconcile positive correlations between recent
price changes and future expected returns for financially distressed firms, they suggest that
the phenomenon becomes less pronounced the further the firm moves away from the default
boundary, where default risk is lower. These implications of the calibration shed light on
the empirical findings of Avramov, Chordia, Jostova, and Philipov (2007) who document
that both the long and the short portfolio of momentum strategies consist of stocks with
high default risk and that momentum profits are restricted to high credit risk firms and
are nonexistent for firms of high credit quality. According to the calibration, distressed
firms are good candidates for portfolio strategies that sort on past returns, since their past
returns are strongly correlated with future expected returns. Moreover, evidence in Bansal,
Dittmar, and Lundblad (2005) supports the theory’s link between risk premia and exposures
to business cycle risks. The authors find that exposures to persistent changes in growth can
explain a large fraction of the variation in expected returns across momentum portfolios.
Economic mechanism. As highlighted above, the firm would always have pro-cyclical
valuations absent leverage, since X(z) is calibrated to be strongly pro-cyclical. Yet, with
leverage and the potential need for external finance, risk is also affected by investors’ financing
and default decisions. These decisions, in turn, depend not only on current beliefs about
the firm’s prospects but also on expectations about the future arrival of information. Even
22
if current beliefs are pessimistic (low π), the firm may be able to attract new funds if
information that helps separate solvent from insolvent firms is expected to arrive in the
near-term. Supporting the firm a little longer is worthwhile in this case as newly arriving
information will help equity holders make better financing decisions than under the current
information set. In particular, information that helps separate good from bad firms allows
equity holders to avoid losses from mistakenly subsidizing insolvent firms and abandoning
solvent firms. Holding current beliefs π fixed, current equity prices are therefore increasing
in the expected future speed of learning.
In the context of the business cycle, the cyclical properties of learning therefore have
first-order effects on the cyclicality of stock valuations. When, in the face of an aggregate
downturn, truly insolvent firms are more likely to exhibit deteriorating earnings performance,
this increases equity values conditional on current beliefs π — ceteris paribus, equity holders
rather quickly observe bad performance by truly insolvent firms, as this signal helps separate
solvent from insolvent firms. This basic benefit of learning implies that, although worse
performance in downturns increases the pro-cyclicality of db-firms’ earnings, it decreases
distressed firms’ equity risk exposures. For low π, risk premia even turn negative as, close to
default, the highlighted learning-based feedback effect becomes a more and more important
determinant of stock prices.
Figure III illustrates these effects based on comparative statics with respect to the parameter λdb,b (B): higher downside risk for insolvent firms in recessions, λdb,b (B), reduces
equity risk premia. Conversely, lower downside risk and a corresponding slower arrival of
information, leads back to the standard result, obtained for example in Merton (1974), that
equity risk premia increase dramatically as the firm approaches default.44 Insolvent firms’
downside risk in recessions, which is associated with dispersion in realized earnings, is thus
an important determinant of risk premia when agents learn about firm solvency. Moreover,
there is strong empirical evidence of rising dispersion and downside risk in recessions.45
44
The reader might wonder why equity risk premia reach a ceiling of 10% in the good aggregate state
of the low parameterization. For beliefs below the threshold π R (B) the firm defaults immediately after a
jump to the recession state B. The corresponding stock return is thus given by −100% for all π < π R (B),
implying a constant risk premium in this region of beliefs.
45
Using micro-level Census data, Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (2014) estimate
a two-state Markov switching model at a quarterly frequency and find that idiosyncratic volatility increases
by 310% in recessions.
23
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FIGURE III
The graphs plot equity risk premia as a function of the posterior probability that the firm is fundamentally solvent, πt = Pr [z = dg|Ft ], and the aggregate state Zt ∈ {G, B}. The graphs illustrate
the dependence of equity risk premia on truly insolvent firms’ downside risk in downturns by solving
the model for different values of λdb,b (B). The labels low, med, and high refer to the parameter values 1.8, 2.5, and 3.2 for λdb,b (B), respectively. All other parameters remain at their baseline values
listed in Table 1. The graph on the left-hand side plots equity risk premia as a function of beliefs
in the boom state G. The right-hand side panel plots equity risk premia in the recession state B.
Downside risk and information. Given that equity holders optimally default after observing a log-decline in earnings of size δdb , the actual parameter value for δdb does not further
affect stock valuations, since equity holders are protected by limited liability. In fact, equity
holders would value the stock in the same way if information from some other source became
available with arrival intensity λdb,b , such as, for example, a negative update to some other
accounting line item. Fundamentally, λdb,b thus represents the arrival intensity of a negative
signal that reveals poor earnings prospects and corresponding insolvency. Comparative statics with respect to λdb,b thus more generally trace out the effects of downside risks emerging
from any source of news about future earnings.
Relation to the business cycle literature. A central prediction of the baseline calibration is that default thresholds are pro-cyclical, that is, the threshold at which default
occurs is lower in downturns, as capital providers are more lenient with respect to current
performance in these times. This prediction is consistent with recent findings of the literature on the cyclical nature of the productivity distribution, in particular Kehrig (2015), who
24
estimates establishment-level productivity levels in the U.S. manufacturing sector from 1972
to 2009 using confidential micro-level Census data. Kehrig’s (2015) empirical results suggest
a pro-cyclical survival cutoff at the bottom tail of the productivity distribution. If failure
is triggered when a firm’s equity value turns zero, this result implies that, as a function of
current productivity, equity valuations close to default are higher in recessions, consistent
with the result of the baseline calibration.
Whereas Kehrig (2015) interprets this empirical finding as evidence of economic factors
that lead to “sullying” recessions, my model reveals that “cleansing” recessions (Schumpeter,
1934) can actually generate this phenomenon as well. Since investors expect to learn about
insolvent firms during the test of an aggregate downturn, they are more willing to support
firms of uncertain quality that currently show mediocre performance46 — a high expected
speed of learning increases the equity value in downturns and warrants more lenience with
respect to current performance. Yet note that this learning-based feedback effect does not
predict fewer defaults in recessions. On the contrary, it is the very fact that equity holders
expect to trigger numerous defaults after more extreme declines that justifies more lenience
with respect to current performance.
Implications for empirical asset pricing tests. When relating the predictions of the
model to the empirical asset pricing literature it is important to note that empirical papers
study expected returns of financially distressed stocks at the portfolio level. Studies such as
Campbell, Hilscher, and Szilagyi (2008) sort the cross-section of stocks based on default likelihood measures and then form portfolios based on breakpoints (e.g., quintile breakpoints).
The degree to which firms in these portfolios are financially distressed varies naturally over
the business cycle: the average firm in the bottom quintile tends to have higher default risk
during a recession than during a boom. Given the calibration’s prediction that firms that are
closer to default have lower exposures to business-cycle risks, this implies that the quintile
portfolio of the most distressed stocks will have lower exposures to these risks in downturns.
Since market risk premia tend to be high in recessions, the model can thus rationalize a nega46
Whereas Schumpeter (1934) was concerned with the general economic benefits of abandoning inefficient
firms in downturns, my paper focuses primarily on equity valuations. If, as discussed at the beginning of
section 2.2, we interpret c as operational cost then equity value is equal to total firm value, and, absent other
frictions, maximizing equity value coincides with maximizing total surplus.
25
tive correlation between market risk premia and portfolio betas at a business cycle frequency.
This negative correlation implies that unconditional CAPM regressions yield positively biased betas and negatively biased alphas,47 helping reconcile the empirical finding that the
portfolio of stocks with the highest default risk generates negative unconditional alphas in
the data.48
While estimating conditional exposures to business-cycle risks should eliminate this misspecification problem, doing so is a challenging task. First, estimating time-varying betas
with high-frequency return data generally leads to biased estimates, as business cycle risks
materialize at low frequencies and since the coexistence of both high-frequency and lowfrequency shocks obfuscates estimation. Second, estimating exposures to business cycle
risks requires long time-series of data, as a business cycle generates essentially only one data
point. The result is low power, even with several decades of return data.
Despite these empirical challenges, existing empirical evidence supports the notion that
cyclical variation in betas can help reconcile negative unconditional distress alphas. Specifically, O’Doherty (2012) finds that the conditional betas of the quintile of stocks that are
most distressed are lower in downturns and are negatively correlated with measures of the
market risk premium.49 O’Doherty (2012) further estimates that this negative correlation
can account for about two thirds of the unconditional distress anomaly alpha. Yet, facing
limited data availability, O’Doherty’s (2012) estimation approach uses daily return data,
which is not ideal.
The role of external finance. In search for a theoretical explanation for cyclical variation
in betas O’Doherty (2012) notes that noisier information about a firm’s underlying asset
values increases equity values and decreases betas when equity is a European call option on
a firm’s final asset payoff, as in Johnson’s (2004) extension of Merton (1974). According to
this argument, noisier information in recessions would help decrease betas in these times, a
prediction that appears to be at odds with the results discussed here. The difference arises
because in a European call setting the firm has no need to raise external finance, implying
47
See, e.g., Grant (1977) and Jagannathan and Wang (1996).
See, e.g., Dichev (1998) and Campbell, Hilscher, and Szilagyi (2008).
49
O’Doherty’s (2012) estimation approach follows Boguth, Carlson, Fisher, and Simutin (2011).
48
26
that there is also no notion of financial distress. Equity holders make only one default
decision at the terminal date, when the final payoff is perfectly observable.50 Prior noisy
information thus cannot harm access to finance or value. The learning-based feedback effect,
which is at the core of my model, is absent.
The two settings also generate markedly different predictions along other dimensions. In
my model, equity risk premia can fall below asset risk premia and even become negative
as the firm’s default probability increases. As a result, momentum dynamics can arise. In
contrast, in the discussed European call setting, equity risk premia are higher than asset risk
premia and increase monotonically with leverage, creating the opposite relation. In addition,
my model is consistent with the empirical finding that the distress anomaly is particularly
strong among net-issuers,51 that is, distressed firms that seek external finance. In contrast,
external finance is absent in the European call setting. Finally, the learning-based feedback
effect of my model can shed light on the role of active investors in facilitating external finance
and affecting risk premia, a feature I will discuss in more detail in the next section.
3.
Externalities from Active Investors
In the following model extension I illustrate how, through the highlighted learning-based
feedback effect, information-gathering “active investors” can create a significant externality
on other investors’ risk exposures and expected returns. The results of this extended model
can shed light on the puzzling empirical finding that regular shareholders not participating in
private placements of public equity (PIPEs) experience post-issue negative abnormal returns
(see, e.g. Hertzel, Lemmon, Linck, and Rees, 2002, Krishnamurthy, Spindt, Subramaniam,
and Woidtke, 2005, Park, 2013).
In the model extension, a private placement of public equity to a large, active investor
with access to superior information generates an externality: using her superior information,
the active investor can better discern solvent from insolvent firms, which improves financing
and liquidation decisions and thereby affects other investors.
50
Default occurs at the terminal date if, at that time, the perfectly observable asset value is below the face
value of debt.
51
See Park (2013).
27
3.1.
The Extended Model
An active investor is endowed with an information production technology that generates
a perfectly precise signal of the hidden firm state z ∈ {db, dg} with Poisson arrival rate a,
where a may depend on the aggregate state Z. The active investor is a financial firm that
maximizes the market value of its portfolio.
Contracting. The analysis considers contracts between management and the active investor that provide the active investor with new equity shares that yield an ownership share
ω at a purchase price κ
¯ . Management acts in the interest of existing shareholders. Contracts
are limited to one-time provisions of new equity, potentially at a price that differs from the
current market price. Management does not renegotiate the contract in the future, which
implies that future equity injections by the active investor occur at regular market prices. In
principle, the transaction with the active investor may occur at any point in time t¯ > 0. Yet
there is no reason to involve the active investor before the firm enters an opaque distressed
state (Ωd ), since the active investor’s information technology is only useful in states in which
the underlying firm-state z is unobservable.
I treat the size of the active investor’s ownership share ω as an exogenous parameter.
In practice, ownership shares may be affected by regulations, as well as by the trade off
between incentives provision on the one hand and potential costs of un-diversification on the
other. Yet, since these forces could coexist without affecting the qualitative results I aim
to illustrate in this section, I abstract from them to keep the model simple and tractable.
Further, as shown below, the chosen parameter value for ω will not affect equity risk premia,
which are the object of interest in this section.
In practice, private investments in public equity (”PIPEs”) are a typical form of distressed equity investments by active investors. Management is directly involved in PIPE
transactions and has the fiduciary duty to act in the interest of existing shareholders. In
these transactions, management typically deliberately offers profits to active investors by
providing them with securities at discounted prices.52
52
See, e.g., Park (2013).
28
Information environment. Changes to the active investor’s ownership in the firm are
assumed to be publicly observable. The resulting lack of “noise” in the system implies
that the active investor cannot extract rents by trading against less informed investors —
all market participants understand that the active investor only trades for informational
reasons. Yet, the active investor can still generate profits, since management is willing to
provide equity to the active investor at a discounted price. Doing so is consistent with
shareholder value maximization, since, once exposed to the firm, the active investor will
have incentives to inform ongoing financing and default decisions. Thus, the active investor
becomes truly “active” after obtaining a stake in a company, as she is involved in evaluating
and influencing firm policies.
I abstract from noise in the system (e.g., noise traders or liquidity shocks to active
investors) for two reasons. First, regulation in the United States requires that investors
acquiring more than 5% of a firm’s equity with the intent to exert control have to file a
schedule 13d with the SEC within 10 days. In addition, investors have to re-file these
forms in case of material changes to their positions (1%). These required regulatory filings
ensure that information about an active investor’s exposure becomes public at a high-enough
frequency to help inform other equity holders’ future financing and default decisions, which
is the central externality channel in the presented setup.
Second, whereas adding noise to the system would be an interesting extension in its
own right, it would be distracting in the context of this section, which aims to shed light
on empirical regularities of buy-and-hold returns after PIPE transactions. A buy-and-hold
strategy does not correspond to random or selective buy-and-sell orders that are typical
features of noisy rational expectations models. By definition, buy-and-hold strategies do
not encompass trades against informed active investors. Finally, adding noise to the system
would also come at the cost of reducing the tractability of the model.
Cash. The firm can hold new funds provided by the active investor as cash that is invested
at the risk-free rate. In case the firm depletes its cash it can attempt to raise additional
funds, just as in the baseline model of section 2. Management can pay out cash as a dividend
at any point in time. In particular, in case of a strategic liquidation, management can pay
out cash before triggering default.
29
The assumption that cash can be paid out as a dividend at any time simplifies the analysis
but is not essential for the highlighted results. If management could not pay out cash, the
firm could naturally only default after depleting the funds provided by the active investor.
At that time, the same optimal default strategies and valuations would apply as in the setup
considered here. The optimal default strategies in the two setups — with and without cash
payouts — are thus closely related.
Bargaining power. Management acting in the interest of existing equity holders always
has the outside option not to issue shares to the active investor. The value of the equity
under this scenario is given by the reservation value V R , characterized in Proposition 2,
which puts an upper bound on the price discount management is willing to offer. How the
surplus from the active investor’s involvement is split between existing shareholders and the
active investor depends on assumptions about bargaining power. Yet, while the distribution
of bargaining power affects the initial purchase price κ
¯ , it does not influence the results for
equity risk premia discussed in the analysis below.53
3.2.
Analysis
The active investor’s information production technology becomes useful as soon as the
firm enters an opaque distressed state Ωd , where investors face uncertainty about firm solvency.54 As discussed in Lemma 1, investors’ beliefs after a jump to an opaque distressed
state are given by π0 (Z). The reservation value is then simply given by the solution for the
equity value under the baseline model, that is, V R (π0 (Zt ) , Zt ). Let V (π, Z, κ
¯ ) denote the
total equity value at the time when the active investor acquires a stake in the firm at price
κ
¯ . The issuance of new equity to the active investor implies a cash injection, generating an
initial excess cash position of κ
¯ . Let Divt denote a discrete payout of excess cash at time
t as a dividend, where 0 ≤ Divt ≤ κt . In opaque distressed states the firm’s cash balance
53
Empirically, the dilution of existing shareholders in PIPE deals suggests that active investors are in fact
able to extract rents (see, e.g., Park, 2013).
54
As there is no uncertainty about solvency in the initial solvent states Ωs , there is also no benefit from
undertaking a PIPE transaction.
30
evolves as follows:

(r (Z)κ + X − c) dt − Div
f
t
d
t
dκt =
0
for κt > 0,
(11)
for κt = 0.
The following proposition characterizes the solution to the active investor’s dynamic decision
problem after acquiring an ω-share of the firm’s equity.
PROPOSITION 3 (Valuations in the presence of an active investor). In opaque distressed
states Ωd the active investor’s value is given by
τ∗
1
ξ (Zτ )
V (πt , Zt , κt ) =
max Et
ω X(zτ ) − c + Divτ dτ
{τ ∗ ,Divτ }
ξ (Zt )
dτ
t
+
= v A (πt , Zt ) + ωκt .
Z
A
(12)
The corresponding Hamilton-Jacobi-Bellman equation yields the following set of ODEs that
the function v A (π, Z) solves for Z ∈ {G, B}:
0 = ω(Xd − c) − rf (Z) v A (π, Z) + vπA (π, Z)θ (Z) (1 − π) π
X
+
πλdg,z0 (Z) (ωV (z 0 , Z) − v A (π, Z))
z 0 6=dg
−
X
(1 − π) λdb,z0 (Z) v A (π, Z)
z 0 6=db
+a(Z) πωV R (dg, Z) − v A (π, Z)
0
+λ (Z, Z 0 ) eφ(Z,Z ) v A (π, Z 0 ) 1{π≥πA (Z 0 )} − v A (π, Z) .
(13)
The active investor optimally proposes a liquidating dividend (for κt > 0) or refuses to inject
additional funds (for κt = 0) immediately after obtaining a bad signal, or when posterior beliefs π fall below an endogenous cutoff value π A (Z). In states Z where θ (Z) < 0 the following
conditions apply: 0 = vπA (π A (Z) , Z) and 0 = v A (π A (Z) , Z). In states Z where θ (Z) > 0,
the following conditions apply: v A (π A (Z) , Z) = 0 and limπ↑1 v A (π, Z) = ωV R (dg, Z), where
V R (dg, Z) is characterized in Appendix A.2. If θ (Z) = 0 then the ODE for state Z simplifies to a nonlinear equation. The firm’s total equity value is given by V (πt , Zt , κt ) =
1 A
V (πt , Zt , κt ). Equity holders optimally agree to default or liquidate the firm when the
ω
31
active investor proposes to do so. Equity holders further choose to default when in aggregate
state Z posterior beliefs π fall below the endogenous cutoff value π A (Z).
Proof. See Appendix A.3.
Informational Externality. Since there are no non-informational reasons to trade, the
active investor will obtain a price of zero when attempting to sell her equity stake, as other
market participants infer that the active investor must have received a negative signal. Thus,
instead, the active investor optimally publicly proposes liquidation after receiving a bad
signal or when beliefs drop below the endogenous default boundaries π A (Zt ). Otherwise,
it is optimal for the active investor to keep her equity share and stay involved. Small
equity holders follow the active investor, as they know that the active investor aims to
maximize shareholder value, given her equity exposure and her inability to make trading
gains. Through this externality channel, equity holders can effectively free ride on the
active investor’s information production after the active investor has obtained a stake in the
company. Due to this ability to free ride, small equity holders have the same optimal default
boundaries as the active investor.
Purchase price. As discussed above, the initial purchase price depends on the distribution
of bargaining power. If the active investor has all the bargaining power, existing shareholders
obtain exactly their reservation value V R , that is, the purchase price κ
¯ is set such that
existing shareholders’ stake is worth V R after the transaction. For an ω-share in the firm’s
post-issuance equity the active investor’s purchase price is then given by:
κ
¯=
V R (π, Z)
− v (π, Z)+ .
(1 − ω)
(14)
In this case, existing shareholders’ ability to free ride after the transaction does not increase
their shareholder value ex ante, as the price discount given to the active investor exactly
offsets the future benefits of free riding. Nonetheless, immediately after the transaction,
shareholders’ risk exposures are affected significantly by the involvement of the active investor, as highlighted in the next subsection.
32
In this context, it is worth highlighting that negative contemporaneous returns for existing
shareholders at PIPE announcements do not provide conclusive evidence that management is
acting against shareholders’ interests.55 By conditioning on firms that issue discounted equity
to an active investor, the econometrician systematically sorts on firms that just received a
negative shock and for that reason approach an active investor. In other words, under the
counter-factual of no active investor involvement, existing equity holders could have lost at
least as much value.
3.3.
A Parameterized Illustration
In this section, I discuss an adjusted parameterization of the model to illustrate how
the involvement of an information-producing active investor affects the equity risk premia
of distressed firms. Throughout, I will focus the analysis on risk premia associated with
the firm’s non-cash assets. To determine the mechanical effect of cash on the overall risk
premium, one simply has to take the value-weighted average of the risk premium on the
firm’s non-cash assets and its cash investments.
I focus on non-cash assets for several reasons. First, I aim to isolate the effects of the
informational channel highlighted in this paper, which does not effect the risk premium on
cash investments. Second, in the model, the firm could pay out cash as a dividend at any
point in time without positively or negatively affecting shareholder value — the size of the
cash position is thus not uniquely determined. Third, in practice, the firm has in principle
some freedom to choose from a range of fairly-priced marketable securities which can carry
different risk premia. Finally, it is worth noting that, in practice, cash positions after PIPE
transactions are not necessarily large, since injected cash is typically used to make lumpy
debt payments, a feature that is not present in my model with continuous debt payments.
Parameter choices. I use the same parameter values as in the baseline calibration listed
in Table 1, except for two adjustments: First, I assume that the active investor’s information
acquisition technology is expected to generate a signal about firm solvency after 3 months,
independent of the aggregate state Z (that is, a(Z) = 4 for all Z). Second, I reduce truly
55
See, e.g., Park (2013) for related empirical evidence.
33
insolvent firms’ downside risk in recessions to λdb,b (B) = 0.8. This parameter change strongly
reduces the pro-cyclicality of truly insolvent firms’ earnings and lowers the speed of learning
from earnings in recessions. I make this adjustment to illustrate how the active investor’s
involvement can still help reduce risk premia when a firm’s earnings are not as informative
in downturns as in the baseline calibration.
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FIGURE IV
The graphs plot equity risk premia as a function of the posterior probability that the firm is
fundamentally solvent, πt = Pr [z = dg|Ft ], the aggregate state Zt ∈ {G, B}, and the involvement of an active investor.
The labels ”With AI” and ”Without AI” refer to the two cases
with and without the involvement of the active investor. The active investor is expected to receive a signal after 3 months (a(Z) = 4, for all Z) and truly insolvent firms’ downside risk
in recessions is given by λdb,b (B) = 0.8. All other parameter values are as listed in Table 1.
Risk premia, alphas, and default boundaries. Figure IV illustrates that, under this
parameterization, equity risk premia increase dramatically close to the default boundaries.
The reason for this change is a slower speed of learning from earnings in downturns, due to
lower downside risk λdb,b (B) relative to the baseline calibration (recall the related comparative
statics in Figure III). Yet Figure IV also reveals that equity risk premia are substantially
lower with the involvement of the active investor.
After the completion of a PIPE transaction that exposes the active investor to the firm,
risk premia thus can decline discontinuously, even if agents had perfectly anticipated the
active investor’s involvement. For example, when the active investor obtains equity at a
discounted price and can extract the additional surplus it generates, equity risk exposures
34
before the transaction are identical to the ones under permanent autarky (that is, without
the involvement of an active investor).56 Yet, as soon as the active investor has received
the discounted equity, risk exposures change discontinuously, reflecting the active investor’s
influence going forward. As a result, estimating betas based on return data that precedes
the transaction lead to biased results, in particular, negative alphas.
With the active investor, default thresholds also become more lenient (that is, π A (Z) <
π R (Z)). Improved access to information increases equity holders’ option value, which justifies
this additional lenience. Importantly, since it is likely that the active investor generates an
informative signal in the near-term, the probability that the firm ever finds itself in the range
of beliefs with elevated risk premia57 is reduced markedly.
Limiting cases and the cyclicality of activism. In fact, in the limiting case where the
active investor is expected to obtain a signal very quickly (a(Z) → ∞ for all Z), equity risk
premia become flat over the whole range of beliefs. Thus, even with very high default risk,
risk premia do not increase, counter to the predictions of standard theories. Strikingly, a
constant speed of learning in this limiting case effectively neutralizes the leverage effect that
usually causes risk premia to rise with default risk. Further, if investor activism is stronger
in recessions (a(B) > a(G)), then learning can induce negative equity risk premia close to
default, even in the considered parameterization where learning from earnings is slow in
downturns. Overall, the active investor can thus boost the value of her stake not only by
increasing expected dividends but also by reducing discount rates.
Empirical predictions. The highlighted results imply a set of unique predictions for the
expected buy-and-hold returns after distressed equity issuances to active investors. Risk
premia are functions of both the involvement and the skill level of the active investor and
generally change discontinuously after a PIPE transaction is completed, even if the transaction is anticipated. These predictions can shed light on the empirical finding that distressed
firms that announce issuances of private placements of public equity have particularly low
56
Equity holders anticipate that, immediately after the transaction, their equity stake will be worth exactly
the reservation value V R , since the active investor can extract the additional equity value it generates.
57
That is, elevated relative to the equity risk premium for π = 1, which is calibrated to be on average
6.33%.
35
estimated abnormal returns going forward.58 The model’s mechanism is further also consistent with the finding that the degree of dilution of existing shareholders is negatively related
to future expected returns.59 If the discounts provided to active investors are reflective of
their skill (that is, their access to information) then discounts and future risk premia should
be related according to the presented theory.
Note that the presence of the active investor only affects risk premia for sufficiently
pessimistic beliefs. For π → 1, risk premia with and without active investor become identical.
Information production by the active investor does not affect equity valuations absent doubts
about firm solvency, as it is not expected to generate a feedback effect on future financing
and default decisions. Yet, when there is sufficient uncertainty about firm solvency, the
active investor’s externality has a large impact on valuations and risk exposures.
Alternative channels through which active investors may affect firm value include proxy
fights, shareholder proposals to replace management, and the alike. Whereas these alternative types of investor activism could be beneficial whenever the firm faces operating decisions,
they do not necessarily relate to firms in financial distress. Further, active investors’ externality is likely particularly significant in the case of financial distress, where the equity value
is almost completely determined by the default option value, which in turn is highly sensitive
to the active investor’s information.
Redistributional effects for cash flows and risk exposures. The active investor’s
information not only creates surplus but also has redistributional effects. In particular, debt
holders of insolvent firms are ceteris paribus worse off, since better informed equity holders
quickly abandon insolvent firms, and thus stop subsidizing debt holders via equity injections.
The analysis shows that although information production can have similarly adverse effects
on debt holders as risk shifting (Jensen and Meckling, 1976), it does not require any change
in the underlying assets. Further, the active investor’s effect on financing and default policies
leads to a reallocation of risk exposures from equity holders to debt holders.
58
See Park (2013) and related evidence in Hertzel, Lemmon, Linck, and Rees (2002) and Krishnamurthy,
Spindt, Subramaniam, and Woidtke (2005).
59
See Park (2013).
36
Robustness: Markov state dynamics & precautionary information production.
As previously discussed in Section 2.2, it is not essential that uncertainty about the underlying firm state arises only in opaque distressed states. The Markov chain setup could be
extended to capture the notion that investors already face uncertainty about the firm’s true
state before the firm lacks sufficient funds to make debt payments. In this case, it would
be useful to involve the active investor as soon as this uncertainty arises, even if equity
holders do not face an immediate default decision at that point in time.60 This is the case,
because precautionary information production by the active investor would become valuable
as soon as there is also uncertainty about solvency (rather than just uncertainty about firm
prospects). Yet, as discussed above, the benefit of this information would naturally increase
the more distressed the firm becomes.
4.
Conclusion
The quality of investors’ information about firm solvency is essential for firm survival in
distress — lacking precise information, investors may refuse to provide new funds to distressed firms and thereby trigger default. In this paper I develop a tractable dynamic model
to shed light on the asset pricing implications of this feedback effect. The model reveals that
learning can rationalize low and even negative expected equity returns for distressed firms,
and that informational externalities from active investors can have first-order effects on risk
premia dynamics. The theory can shed light on a variety of asset pricing puzzles related
to financial distress, such as distressed firms’ apparent stock market underperformance, momentum return dynamics, and the connection between active investors and expected returns
after private placements of public equity.
60
As long as the firm generates positive dividends it is always optimal not to default, independent of the
underlying hidden firm state.
37
A.
A.1.
Proofs
Equity Value in Initial Solvent States
PROPOSITION 4 (Equity value in initial solvent states). In the solvent states (z, Z) ∈ Ωs
the firm’s equity value is given by
Z
V (zt , Zt ) = max
Et
∗
{τ }
τ∗
t
ξ (Zτ )
((X (zτ ) − c) dτ ) ,
ξ (Zt )
(15)
The corresponding Hamilton-Jacobi-Bellman equation implies that the function V (z, Z) solves
the following system of equations for all (z, Z) ∈ Ωs :
0 = X (z) − c − rf (Z) V (z, Z)
X
+
λzz0 (Z) V R (πt˜ (Z) , Z) − V (z, Z)
z 0 ∈{dg,db}
+
X
λzz0 (Z) (V (z 0 , Z) − V (z, Z))
z 0 ∈Ωz \{dg,db}
+
X
0
λ (Z, Z 0 ) eφ(Z,Z ) (V (z, Z 0 ) − V (z, Z)) .
(16)
Z 0 ∈ΩZ
Proof. The result follows immediately from the Hamilton-Jacobi-Bellman equation. The
equity value in opaque distressed states, V R , is characterized in Proposition 2.
A.2.
Proof of Proposition 2
Equity value in good distressed states (dg, Z). It is useful to first characterize the
firm’s equity value in states (dg, Z):
R
Z
V (dg, Zt ) = Et
t
∞
ξ (Zτ )
(X (zτ ) − c) dτ .
ξ (Zt )
38
(17)
The corresponding HJB equation implies that the function V R (dg, Z) solves the following
system of linear equations for all Z:
0 = x − c − rf (Z) V R (dg, Z) +
X
λdg,z0 (Z) V R (z 0 , Z) − V R (dg, Z)
(z 0 ,Z)∈Ωg
φ(Z,Z 0 )
+λ (Z, Z 0 ) e
V R (dg, Z 0 ) − V R (dg, Z) .
(18)
Boundary Conditions. If θ (Z) < 0 in state Z, then v R (πt , Z) satisfies the smooth
pasting and value matching conditions,
0 = θ (Z) · 1 − π R∗ (Z) · π R∗ (Z) · vπR (π R∗ (Z) , Z),
(19)
0 = v R (π R∗ (Z) , Z).
(20)
If θ (Z) > 0, then π R∗ (Z) has to be chosen such that v R (π R∗ (Z) , Z) = 0 and v R (1, Z)
matches V (dg, Z). If θ (Z) = 0, then the ODE for state Z simplifies to a nonlinear equation.
To verify these boundary conditions, let V¯ R πt , Zt , τ R denote the equity value given beliefs
πt , the aggregate state Zt , and given that the agent follows a strategy of abandoning the
firm at time t + τ R if no jump to any other state occurs in the time between t and t + τ R .
The first-order necessary condition for τ R yields
∂ V¯ R πt , Zt , τ R
∂τ R
= 0.
(21)
τ R =τ R∗
A change in variables yields alternatively
!
∂ V˜ R πt , Zt , π R dπ R τ R , πt , Zt ·
∂π R
dτ R
= 0,
(22)
τ R =τ R∗
where I define the function π R τ R , πt , Zt as follows:
π
R
R
τ , πt , Zt =
R
1 − πt
1 + e(−τZt ·θ(Zt )) ·
πt
39
−1
,
(23)
implying that for all πt ∈ (0, 1) we have
necessary condition may also be written as
dπ R
dτ R
∂ V˜ R πt , Zt , π R
∂π R
> 0. Thus, for πt ∈ (0, 1) the first-order
= 0,
(24)
π R =π R∗ (Zt )
where I define
π R∗ (Zt ) ≡ π R τ R∗ , πt , Zt .
(25)
Notice that for θ (Z) > 0, any τ R ≥ 0 will correspond to π R ≥ πt , since waiting time τ R
increases the conditional probability πt . Let V R (πt , Zt ) denote the value function from the
optimal solution of the equity holders’ problem. Given the assumption that the equity value
is strictly positive at πt , i.e. V R (πt , Zt ) > 0, we obtain
V R π R τ R , πt , Zt , Zt ≥ V R (πt , Zt ) > 0, for all τ R ≥ 0,
(26)
R (π,Z )
t
≥ 0 for all π ∈ [πt , 1], since π R τ R , πt , Zt ≥ πt for all τ R ≥ 0.
given that ∂V ∂π
∂V R (π,Zt )
≥ 0 follows from the assumption that V (dg, Zt ) > V (db, Zt ) = 0. Thus, given
∂π
that V R (πt , Zt ) > 0 and θ (Z) > 0, the equity value after any positive waiting time is also
positive, and thus it is optimal not to abandon the firm as long as it stays in the current
state, that is, it is optimal to set τ R∗ = ∞. Since
lim π R τ R , πt , Zt = 1,
τ R →∞
(27)
the ODE simplifies in the limit τ R → ∞ to the non-linear equation that the function
V (dg, Zt ) solves, implying that
lim V R (πt , Zt ) = V (dg, Zt ) .
π→1
R
(28)
< 0, implying that waiting time
On the other hand, for θ (Z) < 0, it follows that dπ
dτ R
corresponds to lower conditional probabilities πt . By assumption, at π = 0 the equity
value is zero (V (db, Z) = 0), and the firm is abandoned. Further, by assumption, we have
40
V R (1, Z) = V (dg, Z) > 0. It is optimal to abandon at π R∗ where π R∗ satisfies
V R π R∗ , Z = 0,
(29)
and where the smooth pasting condition
∂V R (π, Zt ) = 0,
∂π
πt =π R∗
(30)
is satisfied. If smooth pasting was not satisfied then there could be an optimal cutoff π Rˆ
where the resulting value function V Rˆ satisfies V Rˆ π Rˆ , Z = 0 and
∂V Rˆ (π, Zt ) > 0.
∂π
πt =π Rˆ
(31)
Yet, then heuristically, at πt = π Rˆ , the agent benefits from waiting another instant ∆t and
abandoning the firm afterwards, since the expected income flow is positive:
(Xd − c +
X
π(t) · λdg,z0 (Z) · V (z 0 , Z)) · ∆t
z 0 ∈Ωg
0
+
+λ (Z, Z 0 ) · eφ(Z,Z ) · v R (πt , Z 0 ) · ∆t
dπt
· ∆t
= −VπRˆ (πt , Z)
dt
> 0.
This contradicts that π Rˆ is an optimal cutoff. Since
R
(32)
∂V Rˆ (π,Zt ) ∂π
πt =π Rˆ
> 0 violates optimiza-
∂V (π,Zt )
tion and since
is weakly positive over the whole domain π ∈ [0, 1], it follows that
∂π
∂V Rˆ (π,Zt ) = 0 must hold at the optimal cutoff π R∗ (Z), given that θ (Z) < 0.
∂π
R∗
πt =π
A.3.
(Z)
Proof of Proposition 3
Learning & passage time. The active investor’s signal arrival rate a(Z) is independent
of the underlying hidden firm state (dg vs. dg) and thus does not alter the evolution of
beliefs absent signals (that is, θ (Z) is identical to the case without an active investor).
41
Excess cash. The excess cash position enters the value function linearly since the firm is
free to pay out cash at any point in time, and since cash is invested at the risk-free rate,
which is value-neutral.
Informational externality. Since trades by the active investor are effectively publicly
observable and cannot generate profits, the active investor maximizes the value of her ωshare of the firm’s equity by maximizing the total equity value. The active investor optimally
immediately reveals information to the management and other shareholders. As the active
investor has no incentive to propose policies that are not shareholder value maximizing, small
investors optimally follow the active investor’s proposals. This ensures that, after bad news
the firm is liquidated, and after good news, the firm is kept afloat, which is in the interest
of both the active investor and small investors. As a result, small shareholders’ maximized
objective is effectively a scaled version of the active investors’ maximized objective, where
the scaling constant is 1/ω. Specifically, with an active investor, the firm’s total equity value
in opaque distressed states is given by
Z
τ∗
ξ (Zτ )
V (πt , Zt , κt ) =
max
E
t
{τ ∗ ,Divτ }
ξ (Zt )
t
+
= v (πt , Zt ) + ωκt .
1
X(zτ ) − c + Divτ dτ
dτ
(33)
The corresponding Hamilton-Jacobi-Bellman equation yields the following set of ODEs that
the function v (π, Z) solves:
0 = Xd − c − rf (Z) v (π, Z) + vπ (π, Z)θ (Z) (1 − π) π
X
+
πλdg,z0 (Z) (V (z 0 , Z) − v (π, Z))
z 0 6=dg
−
X
(1 − π) λdb,z0 (Z) v (π, Z)
z 0 6=db
+a(Z) πV R (dg, Z) − v(π, Z)
0
+λ (Z, Z 0 ) eφ(Z,Z ) v (π, Z 0 ) 1{π≥πA (Z 0 )} − v (π, Z) .
(34)
Given the above statements, the derivation of the boundary conditions is analogous to the
proof of Proposition 2 in Appendix A.2.
42
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